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| Mirrors > Home > MPE Home > Th. List > sbalex | Structured version Visualization version GIF version | ||
| Description: Equivalence of two ways
to express proper substitution of a setvar for
another setvar disjoint from it in a formula. This proof of their
equivalence does not use df-sb 2065.
That both sides of the biconditional express proper substitution is proved by sb5 2276 and sb6 2085. The implication "to the left" is equs4v 1999 and does not require ax-10 2141 nor ax-12 2177. It also holds without disjoint variable condition if we allow more axioms (see equs4 2421). Theorem 6.2 of [Quine] p. 40. Theorem equs5 2465 replaces the disjoint variable condition with a distinctor antecedent. Theorem equs45f 2464 replaces the disjoint variable condition on 𝑥, 𝑡 with the nonfreeness hypothesis of 𝑡 in 𝜑. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2268 in place of equsex 2423 in order to remove dependency on ax-13 2377. (Revised by BJ, 20-Dec-2020.) Revise to remove dependency on df-sb 2065. (Revised by BJ, 21-Sep-2024.) (Proof shortened by SN, 14-Aug-2025.) |
| Ref | Expression |
|---|---|
| sbalex | ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfe1 2150 | . . 3 ⊢ Ⅎ𝑥∃𝑥(𝑥 = 𝑡 ∧ 𝜑) | |
| 2 | ax12ev2 2180 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → (𝑥 = 𝑡 → 𝜑)) | |
| 3 | 1, 2 | alrimi 2213 | . 2 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
| 4 | equs4v 1999 | . 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → ∃𝑥(𝑥 = 𝑡 ∧ 𝜑)) | |
| 5 | 3, 4 | impbii 209 | 1 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: equsexvOLD 2269 sb5 2276 dfsb7 2279 alexeqg 3651 dfdif3OLD 4118 pm13.196a 44433 |
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